Honor Among Thieves

There is an ancient idea in philosophy that a large part of ethics can be
explained or justified as enlightened self-interest for social animals, who
must live with and depend upon each other. One way this was expressed was the
enduring common-place that even a group of bandits must show some
justice to one another, if they are to successfully rob others. For two
centuries now, lots of economics has been motivated by trying to show that
peaceful haggling in the market place is better for everyone than less
cooperative struggles, or conversely by skepticism that everything is really so
providentially arranged. More recently, since the 1970s a loose community of
theoretical biologists, economists, political scientists, mathematicians and
philosophers has tried to understand how cooperation, norms, and even altruism
might appear among purely selfish agents, using the tools of evolutionary game
theory. (The goal is to prove that honesty is the best policy, and virtue a
dominant strategy.) Karl Sigmund has been a leading figure in these efforts,
and The Calculus of Selfishness is his latest attempt at an
introduction to the field. In its exposition, the book focuses on reciprocity
between pairs of self-interested individuals in certain elementary types of
interactions.

Game theory (as laid out in chapter 1) models agents interacting with each
other, in pairs or in larger groups, with a fixed set of moves available to
them. (Sigmund mostly deals with two-player games, but this is just for
simplicity.) An agent — Alice, say — has a "strategy" that tells
her what move to make at each step in the game, in response to another agent's
moves and to the state of the external world (if the model admits that the
latter exists). Alice's coplayer — Bob, say — also has a strategy,
and together the two strategies determine the outcome of the game. At its end,
Alice and Bob each get a "payoff", according to a function that depends on the
moves both have made. Their strategies are in equilibrium if neither of them
could increase their payoff by changing their moves unilaterally.

Basic economics courses lead one to expect that there will be only one
equilibrium and that it will be optimal for everyone. The games of relevance to
the evolution of cooperation, however, are "social dilemmas", where this
expectation fails. Sometimes the problem is that the equilibrium is optimal for
no one. Imagine that Alice and Bob are two bandits, who can either "cooperate"
in robbing villages and caravans, or "defect" by turning on each other. If they
both cooperate, each will take $1,000; if they both defect, neither can steal
effectively and they'll get $0. But suppose that if Alice cooperates and Bob
defects by turning on her, he will get $2,000 and she will lose $500 —
and vice versa. Then regardless of what Alice does, Bob will be better off
defecting. So by symmetry, the only way to achieve equilibrium is for both of
them to defect — even though they'd both be better off, in purely selfish
terms, if they both cooperated. When this type of problem was first posed at
the RAND Corporation in the early 1950s, it was framed as prisoners being
offered the chance to turn state's witness, so it is still called the
"prisoners' dilemma", and it is the archetype of many cooperation problems. A
multiperson version of it is the public goods game, in which something
like an open park is shared equally among all participants, no matter how much
each of them has contributed to its creation or upkeep.

In other social dilemmas, there are many equilibria, some more favorable to
one player than to another. If Alice and Bob can agree to cooperate as bandits,
how should they split the loot — the extra loot, that is, beyond what
they would each get on their own? If Alice can convince Bob that she won't
cooperate unless she gets, say, 75 percent of the gains, it is in Bob's
interest to agree to her demand, since 25 percent extra is better than 0
percent. Thus there is an Alice-favoring equilibrium; but, symmetrically, there
is also a Bob-favoring one. However, agreeing to a 50-50 split is not an
equilibrium at all. This is called the "snowdrift game", after another fable,
and is the archetypal depiction of the inevitable struggle over how to divide
the gains from cooperation.

In evolutionary game theory (introduced in chapter 2), we imagine a
population of agents, all with their own strategies, who are randomly paired up
to play a given game. The "replicator dynamic" means that strategies whose
players do better than average will increase their share of the population, at
the expense of strategies that have worse-than-average results. Whether this
works by agents imitating the more successful among them, or by actual natural
selection, is not (at this level of abstraction) mathematically relevant. What
matters is that the performance of each strategy depends on what other
strategies are available in the population and how common they are. The
population-level dynamics of strategies are stochastic processes, which in the
large-population limit (where Sigmund mostly works) smooth out into
differential equations.

When are cooperative strategies able to survive or even dominate the
population? Chapters 3, 4 and 5 consider mechanisms that can achieve
this. Chapter 3 is about direct reciprocity: If Alice and Bob play the same
game with each other repeatedly, and each can remember what the other player
did earlier, they can pay each other back in kind. In games of the prisoner's
dilemma type, a strategy of cooperating if and only if the other player
cooperated last time does not do well against a fixed strategy of
noncooperation, but it does do well against another player with the
same strategy. It can thus invade a population of noncooperators, although once
it becomes dominant, a strategy of unconditional cooperation can invade in
turn. These strategies require only very basic sorts of memory; more complex
strategies, relying on more sophisticated memories, can be more robust to
noise, avoiding costly cycles of retaliation provoked by error and
misunderstanding.

Chapter 4 extends these ideas to indirect reciprocity, in two forms. In
both, Alice's decision to cooperate with Bob today leads to cooperation
tomorrow, involving a third party. In one form, today's cooperation leads
Charlie to cooperate with Alice tomorrow; in the other, it leads Bob to
cooperate with Charlie.

Chapter 5 looks at norms of fairness and the effects of having a reputation
for refusing to violate norms. In the split-the-loot game above, if Alice has
established a reputation for refusing to cooperate at all rather than accept
unfair divisions, Bob will not bother to offer her a bad deal in the first
place. Both indirect reciprocity and reputation require not just memory but
also fairly accurate information about what third parties have done —
information which must be acquired by perception or communication.

Chapter 6 looks in detail at a particular model of the provision of public
goods, and at whether voluntary or compulsory participation in providing public
goods works better. This chapter is exemplary as an analysis of a model, but
the conclusions seem to rest heavily on the assumption that each agent's reward
from the public good goes down in proportion to the population size, which is
not true of many public goods, such as roads free of bandits.

Up to this point, the book has assumed that players meet randomly, which is
of course unrealistic. The final chapter looks briefly at what happens when
some internal structure is added to the populations, by spreading them out in
space and limiting agents to playing with their immediate neighbors, or by
dividing the population into subgroups that mix internally and periodically
re-form. This chapter feels less complete than the rest, because it offers just
a glimpse of a vast, largely unmapped territory.

Sigmund's mathematical exposition is exemplary. He starts with the
presumption that the reader has only rudimentary linear algebra (multiplying
vectors by matrices) and some notion of what a differential equation is, and he
builds up from there, introducing more advanced concepts and results as
needed. He avoids formal proofs and bookkeeping in favor of careful
explanations of key points and illustrative calculations. As he teaches
evolutionary game theory, Sigmund is also demonstrating how to write about
applied mathematics.

Of course this is not just a mathematical book. That Sigmund calls his two
forms of indirect reciprocity "vicarious" and "misguided", rather than, for
instance, "admiring" and "grateful", doesn't change his models, but it probably
does shape how readers interpret the models and their properties, and the
interpretations are what give the models their interest. (Would anyone study
these equations for their own sake?) Since Sigmund, like everyone else in the
field, is trying to explain how ethics could arise from selfishness, I suspect
that morally charged language is unavoidable, and even desirable, if taken with
a grain of salt. These are not matters that mathematics could ever settle, but
it is important, and surprising, that mathematics can illuminate them. And
Sigmund's book casts a very bright light.